Subsequences of a Multiset

We call a partition \(\mu = (\mu_1, \ldots, \mu_k)\) of \(m\), \(m \leq n\), a constrained induced partition (cip) from a partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) of \(n\) if \(\mu_i \leq \lambda_i\) for \(i = 1, 2, \ldots, k\). In this paper, we study the set of cips (Sections 1-2), determine cips of size \(p\) (Section 4), and give a formula for the number of total subsequences with fixed size chosen from a given multiset such that the multiplicity of each digit in a subsequence is less than or equal to the multiplicity of this digit in the given multiset.